A294243 Sum of the larger parts of the partitions of 2n into two parts with smaller part nonsquarefree.

0, 0, 0, 4, 6, 8, 10, 20, 33, 39, 45, 63, 71, 79, 87, 111, 121, 149, 161, 193, 207, 221, 235, 273, 314, 332, 377, 425, 447, 469, 491, 545, 569, 593, 617, 677, 703, 729, 755, 821, 849, 877, 905, 977, 1052, 1084, 1116, 1196, 1279, 1365, 1403, 1493, 1533, 1627

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..n} (2*n - i) * (1 - mu(i)^2), where mu is the Möbius function (A008683).

Maple

N:= 100: # to get a(1)..a(N)
S:= ListTools:-PartialSums(map(t -> `if`(numtheory:-issqrfree(t), [0, 0], [1, t]), [$1..N])):
seq(2*n*S[n, 1]-S[n, 2], n=1..N); # Robert Israel, Oct 27 2017

Mathematica

Table[Sum[(2 n - k) (1 - MoebiusMu[k]^2), {k, n}], {n, 80}]
Table[Total[Select[IntegerPartitions[2 n, {2}], !SquareFreeQ[#[[2]]]&][[;; , 1]]], {n, 60}] (* Harvey P. Dale, Apr 09 2023 *)

Prog

(PARI) a(n) = sum(i=1, n, (2*n-i)*(1-moebius(i)^2)); \\ Michel Marcus, Oct 27 2017

Crossrefs

Cf. A008683, A008966, A294244.

Sequence in context: A289424 A050835 A369139 * A349694 A374784 A298473

Adjacent sequences: A294240 A294241 A294242 * A294244 A294245 A294246

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 25 2017

Status

approved